Percolation on fitness-dependent networks with heterogeneous resilience

Hoppe, K.; Rodgers, G. J.

doi:10.1103/physreve.90.012815

“…The beauty of percolation [5] lays both in its simplicity and possible practical applications. The latter ranges from theoretical studies of geometrical model of the phase transition [6], via condensed matter physics [7], rheology [8], forest fires [9] to immunology [10] and quantum mechanics [11].…”

Section: Introductionmentioning

confidence: 99%

Simple cubic random-site percolation thresholds for neighborhoods containing fourth-nearest neighbors

Malarz

¹

2015

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In the paper random-site percolation thresholds for simple cubic lattice with sites' neighborhoods containing next-next-next-nearest neighbors (4NN) are evaluated with Monte Carlo simulations. A recently proposed algorithm with low sampling for percolation thresholds estimation [Bastas et al., arXiv:1411.5834] is implemented for the studies of the top-bottom wrapping probability.The obtained percolation thresholds are pC ( (12), where 3NN, 2NN, NN stands for next-nextnearest neighbors, next-nearest neighbors, and nearest neighbors, respectively. As an SC lattice with 4NN neighbors may be mapped onto two independent interpenetrated SC lattices but with two times larger lattice constant the percolation threshold pC(4NN) is exactly equal to pC(NN). The simplified Bastas et al. method allows for reaching uncertainty of the percolation threshold value pC similar to those obtained with classical method but ten times faster.

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“…If we imagine links are added one at a time at a given rate, from the kernel f (x, y) we can derive the probability that a node with fitness x increases its degree by one as [45] λ…”

Section: Model Setupmentioning

confidence: 99%

“…We define the average fraction f + = N + /N . The network constructed via the sequential deposition of links (as described above) may undergo a percolation transition [44,45,49,50] as a function of f + , such thatbeyond a critical value of f + -a giant connected component of N C nodes emerges, whose fractional average size S = N C /N remains finite as N → ∞. We stress that in any fixed instance N C ≤ N + , since some high-fitness nodes may still not engage in LN (see Fig.…”

Section: Model Setupmentioning

confidence: 99%

“…We model the emergence of the Lightning Network as a (bond) percolation process on a graph, exploring how different conditions may impact its feasibility [44]. In particular, we consider fitness-dependent network models [45][46][47][48] where the probability of creating a new edge depends on intrinsic node features collectively denoted node fitness. In the LN case, the node fitness will be defined in terms of the node wealth and activity (i.e.…”

Section: Introductionmentioning

confidence: 99%

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A Percolation Model for the Emergence of the Bitcoin Lightning Network

Bartolucci

¹

,

Caccioli

²

,

Vivo

³

2019

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The Lightning Network is a so-called second-layer technology built on top of the Bitcoin blockchain to provide "off-chain" fast payment channels between users, which means that not all transactions are settled and stored on the main blockchain. In this paper, we model the emergence of the Lightning Network as a (bond) percolation process and we explore how the distributional properties of the volume and size of transactions per user may impact its feasibility. The agents are all able to reciprocally transfer Bitcoins using the main blockchain and also -if economically convenientto open a channel on the Lightning Network and transact "off chain". We base our approach on fitness-dependent network models: as in real life, a Lightning channel is opened with a probability that depends on the "fitness" of the concurring nodes, which in turn depends on wealth and volume of transactions. The emergence of a connected component is studied numerically and analytically as a function of the parameters, and the phase transition separating regions in the phase space where the Lightning Network is sustainable or not is elucidated. We characterize the phase diagram determining the minimal volume of transactions that would make the Lightning Network sustainable for a given level of fees or, alternatively, the maximal cost the Lightning ecosystem may impose for a given average volume of transactions. The model includes parameters that could be in principle estimated from publicly available data once the evolution of the Lighting Network will have reached a stationary operable state, and is fairly robust against different choices of the distributions of parameters and fitness kernels.

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“…We model the emergence of the Lightning Network as a (bond) percolation process on a graph, exploring how different conditions may impact its feasibility 46 . In particular, we consider fitness-dependent network models [47][48][49][50] where the probability of creating a new edge depends on intrinsic node features collectively denoted node fitness. In the LN case, the node fitness will be defined in terms of the node wealth and activity (i.e.…”

mentioning

confidence: 99%

A percolation model for the emergence of the Bitcoin Lightning Network

Bartolucci

¹

,

Caccioli

²

,

Vivo

³

2020

Sci Rep

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The Lightning Network is a so-called second-layer technology built on top of the Bitcoin blockchain to provide "off-chain" fast payment channels between users, which means that not all transactions are settled and stored on the main blockchain. In this paper, we model the emergence of the Lightning Network as a (bond) percolation process and we explore how the distributional properties of the volume and size of transactions per user may impact its feasibility. The agents are all able to reciprocally transfer Bitcoins using the main blockchain and also -if economically convenient -to open a channel on the Lightning Network and transact "off chain". We base our approach on fitness-dependent network models: as in real life, a Lightning channel is opened with a probability that depends on the "fitness" of the concurring nodes, which in turn depends on wealth and volume of transactions. The emergence of a connected component is studied numerically and analytically as a function of the parameters, and the phase transition separating regions in the phase space where the Lightning Network is sustainable or not is elucidated. We characterize the phase diagram determining the minimal volume of transactions that would make the Lightning Network sustainable for a given level of fees or, alternatively, the maximal cost the Lightning ecosystem may impose for a given average volume of transactions. The model includes parameters that could be in principle estimated from publicly available data once the evolution of the Lighting Network will have reached a stationary operable state, and is fairly robust against different choices of the distributions of parameters and fitness kernels.Bitcoin, the pioneering cryptocurrency, has brought about an unprecedented revolution in the payment industry 1 . Despite its traction and success over the last ten years, the original blockchain -the technological infrastructure underlying Bitcoin -suffers from some limitations that may hinder the future growth and adoption of the cryptocurrency. One of the major issues is the scalability of the system: the current number of transactions validated via this platform is between 3 and 7 transactions per second, compared for instance to thousands of transactions handled by the Visa circuit 2 . The lack of scalability is mainly caused by constraints on throughput of transactions, with the block size fixed at 1MB, and by the high latency -with a new block created on average only every ten minutes. Those limitations are imposed to safeguard the security of the platform against malicious attacks and are difficult to relax without major changes in the protocol.The main solutions proposed to address the scalability issue include (i) changes to the main protocol (consensus algorithm, parameters) and (ii) sidechains and second-layer solutions (see 3 for a recent technical review). Notable examples of type-(i) solutions include new consensus protocols, which would allow a faster issuance of new blocks among other new features 4 . Sidechains are blockchains "conne...

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Percolation on fitness-dependent networks with heterogeneous resilience

Cited by 7 publications

References 33 publications

Simple cubic random-site percolation thresholds for neighborhoods containing fourth-nearest neighbors

Simple cubic random-site percolation thresholds for neighborhoods containing fourth-nearest neighbors

A Percolation Model for the Emergence of the Bitcoin Lightning Network

A percolation model for the emergence of the Bitcoin Lightning Network

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